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Answer :
Sure! Let's solve this problem by completing the square.
The given quadratic equation is:
[tex]\[ x^2 - 10x - 81 = 0 \][/tex]
To solve this equation by completing the square, follow these steps:
1. Move the constant term to the other side:
Start by isolating the terms involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ x^2 - 10x = 81 \][/tex]
2. Complete the square:
To complete the square, you'll need to add a number to both sides of the equation to make the left side a perfect square trinomial. This number is found by taking half of the coefficient of [tex]\( x \)[/tex], squaring it, and adding it to both sides.
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-10\)[/tex].
- Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex].
- Squaring [tex]\(-5\)[/tex] gives [tex]\(25\)[/tex].
Add [tex]\(25\)[/tex] to both sides of the equation:
[tex]\[ x^2 - 10x + 25 = 81 + 25 \][/tex]
Now, the equation becomes:
[tex]\[ (x - 5)^2 = 106 \][/tex]
This is step 4 from the provided options.
3. Correct option:
The correct step that completes the square correctly as shown is:
[tex]\[ x^2 - 10x + 25= 81 + 25 \][/tex]
This step forms a perfect square on the left-hand side, which helps us solve the equation more easily. Therefore, the correct option is step 4.
The given quadratic equation is:
[tex]\[ x^2 - 10x - 81 = 0 \][/tex]
To solve this equation by completing the square, follow these steps:
1. Move the constant term to the other side:
Start by isolating the terms involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ x^2 - 10x = 81 \][/tex]
2. Complete the square:
To complete the square, you'll need to add a number to both sides of the equation to make the left side a perfect square trinomial. This number is found by taking half of the coefficient of [tex]\( x \)[/tex], squaring it, and adding it to both sides.
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-10\)[/tex].
- Half of [tex]\(-10\)[/tex] is [tex]\(-5\)[/tex].
- Squaring [tex]\(-5\)[/tex] gives [tex]\(25\)[/tex].
Add [tex]\(25\)[/tex] to both sides of the equation:
[tex]\[ x^2 - 10x + 25 = 81 + 25 \][/tex]
Now, the equation becomes:
[tex]\[ (x - 5)^2 = 106 \][/tex]
This is step 4 from the provided options.
3. Correct option:
The correct step that completes the square correctly as shown is:
[tex]\[ x^2 - 10x + 25= 81 + 25 \][/tex]
This step forms a perfect square on the left-hand side, which helps us solve the equation more easily. Therefore, the correct option is step 4.
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